Numerical simulation of two-dimensional combustion using mesh-free methods

The purpose of this research was to develop tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs). Mesh-free methods offer many distinct advantages over traditional finite difference, finite element, and finite volume methods. Traditional Lagrangian methods with significant swirl require mesh stiffeners and periodic remeshing to avoid excessive mesh distortion; such codes often require user interaction to repair the meshes before the simulation can proceed again. A propagating flame of infinite extent is simulated as a collection of normalized cells with periodic boundary conditions. Rather than capturing the flame front, it is tracked as a discontinuity. The flame front is approximated as a product of a Heaviside function in the normal propagation direction and a piece-wise continuous function represented by RBFs in the tangential direction. The cells are subdivided into the burned and unburned sub-domains approximated by two-dimensional periodic RBFs that are constrained to be strictly conservative. The underlying steady flow is vortical with an input turbulent intensity. The governing equations are rotationally and translationally transformed to produce exact differentials that are integrated exactly in time. In the present paper, the previous results of Aldredge who used a finite-difference level-set method were compared. The physical behavior was remarkably similar, whereas the finite-difference level-set method required 14 h of CPU time, the RBF approach required only 120 CPU seconds on a desktop computer for the case with the largest turbulent intensity. Although there are no other papers that tried to duplicate the original results of Aldredge, the results that are reported here are consistent with the physics observed in other experimental and numerical investigations.